Endo M., Iijima S., Dresselhaus M.S. (eds.) Carbon nanotubes
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NOTE ADDED
IN
PROOF
Since this paper was written, low-temperature mea-
surements on carbon nanotubes revealed the existence
of Universal Conductance Fluctuations with magnetic
field. These results will be reported elsewhere.
L.
Langer,
L.
Stockman,
J.
P.
Heremans,
V.
Bayot, C.
H.
Olk,
C.
Van
Haesendonck, Y. Bruynseraede, and
J.
P.
Issi, to be published.
VIBRATIONAL MODES
OF
CARBON NANOTUBES;
SPECTROSCOPY AND THEORY
P.
C.
EKLUND,’
J.
M.
HOLDEN,’
and
R.
A.
JISHI*
’Department of Physics and Astronomy and Center for Applied Energy Research,
University of Kentucky, Lexington,
KY
40506,
U.S.A.
’Department of Physics, Massachusetts Institute of Technology, Cambridge,
MA
02139, U.S.A.;
Department of Physics, California State University,
Los
Angeles, CA 90032,
USA.
(Received
9
February
1995;
accepted
in
revised
form
21
February
1995)
Abstract-Experimental and theoretical studies of the vibrational modes of carbon nanotubes are reviewed.
The closing of
a
2D graphene sheet into
a
tubule is found to lead to several new infrared (1R)- and Raman-
active modes. The number
of
these modes
is
found to depend
on
the tubule symmetry and
not
on
the di-
ameter. Their diameter-dependent frequencies are calculated using
a
zone-folding model. Results of Raman
scattering studies on arc-derived carbons containing nested or single-wall nanotubes are discussed. They
are compared
to
theory and to that observed for other
sp2
carbons also present in the sample.
Key
Words-Vibrations, infrared, Raman, disordered carbons, carbon nanotubes, normal modes.
1. INTRODUCTION
In this paper, we review progress in the experimental
detection and theoretical modeling
of
the normal modes
of
vibration
of
carbon nanotubes. Insofar as the theo-
retical calculations are concerned, a carbon nanotube
is assumed to be an infinitely long cylinder with a mono-
layer of hexagonally ordered carbon atoms in the tube
wall.
A
carbon nanotube is, therefore,
a
one-dimensional
system in which the cyclic boundary condition around
the tube wall, as well as the periodic structure along
the tube
axis,
determine the degeneracies and symmetry
classes of the one-dimensional vibrational branches
i1-31
and the eIectronic energy bands[4-12].
Nanotube samples synthesized in the laboratory are
typically not this perfect, which has led
to
some confu-
sion in the interpretation of the experimental vibrational
spectra. Unfortunately, other carbonaceous material
(e.g., graphitic carbons, carbon nanoparticles, and
amorphous carbon coatings on the tubules) are also
generally present in the samples, and this material
may contribute artifacts to the vibrational spectrum.
Defects in ithe wall (e.g., the inclusion of pentagons
and heptagons) should also lead to disorder-induced
features in the spectra. Samples containing concentric,
coaxial, “nested” nanotubes with inner diameters
-8
nm and outer diameters
-80
nm have been
syn-
thesized using carbon arc methods[l3,14], combustion
fllames[l5], and using small
Ni
or
Co
catalytic parti-
cles in hydrocarbon vapors[lb-201. Single-wall nano-
tubes (diameter 1-2 nm) have been synthesized by
adding metal catalysts
to
the carbon electrodes in
a
dc
arc[21,22]
~
To
date, several Raman scattering stud-
ies[23-281 of nested and single-wall carbon nanotube
samples have appeared.
2.
OVERVIEW
OF
RAMAN SCATTERING
FROM
SP2
CARBONS
Because
a
single carbon nanotube may be thought
of
as
a
graphene sheet rolled up to form
a
tube, car-
bon nanotubes should be expected
to
have many prop-
erties derived from the energy bands and lattice
dynamics of graphite. For the very smallest tubule di-
ameters, however, one might anticipate new effects
stemming from the curvature
of
the tube wall and the
closing of the graphene sheet into a cylinder.
A
natu-
ral starting point for the discussion of the vibrational
modes
of
carbon nanotubes is, therefore, an overview
of
the vibrational properties
of
sp2
carbons, includ-
ing carbon nanoparticles, disordered
sp2
carbon, and
graphite.
This
is also important because these forms
of carbon are also often present in tubule samples as
“impurity phases.”
In Fig. la, the phonon dispersion relations for 3D
graphite calculated from
a
Born-von Karman lattice-
dynamical model are plotted along the high symmetry
directions of the Brillouin zone (BZ). For comparison,
we show, in Fig. lb, the results of a similar calcula-
tion[29] for a 2D infinite graphene sheet. Interactions
up
to
fourth nearest neighbors were considered, and
the force constants were adjusted
to
fit
relevant exper-
imental data in both
of
these calculations. Note that
there is little dispersion in the
k,
(I?
to
A)
direction
due
to
the weak interplanar interaction in
3D
graphite
(Fig. IC).
To
the right of each dispersion plot
is
the
calculated one-phonon density
of
states. On the energy
scale of these plots, very little difference
is
detected
between the structure of the
2D
and 3D one-phonon
density of states. This is due to the weak interplanar
coupling in graphite. The eigenvectors for the optically
129
130
P.
C.
EKLUND
et
al.
M
K
t
kz
Wave
Vector
(a)
IK
r
wovevecior
42
cm-1
ro
0,5
1
,o
g
(w)(
I
o-"
states/crn3cm-l)
Ramon
active
1582
cm-1
E2s*
868 cm-1
lnfrare
1588 cm-1
act we
Went
870 cm-1
(d
(dl
Fig.
1.
Phonon
modes in
2D
and
3D
graphite:
(a)
3D
phonon dispersion,
(b)
2D
phonon dispersion,
(c)
3D
Brillouin zone, (d) zone center
q
=
0
modes
for
3D
graphite.
Vibrational
modes
of
carbon
nanotubes
131
allowed r-point vibrations for graphite
(3D)
are shown
in Fig. Id, which consist
of
two, doubly degenerate,
Raman-active modes
(E;;)
at
42
cm-',
E;:'
at
1582
cm-I), a doubly degenerate, infrared-active
El
,
mode
at
1588
cm-'
,
a nondegenerate, infrared-active
AZu
mode at
868
cm-', and two doubly degenerate
Bzg
modes
(127
cm-',
870
cm-') that are neither Raman-
nor infrared-active. The lower frequency
Bii)
mode
has been observed by neutron scattering, and the other
is predicted at
870
cm-'. Note the I'-point
El,
and
15;;)
modes have the same intralayer motion, but dif-
fer in the relative phase of their C-atom displacements
in
adjacent layers. Thus, it is seen that the interlayer
interaction in graphite induces only an
-6
cm-' split-
ting between these modes
(w(El,)
-
@(Ez,
)
=
6
cm-')). Furthermore, the frequency of the rigid-layer,
shear mode
(o(E2;))
=
42 cm-') provides a second
spectroscopic measure of the interlayer interaction be-
cause, in the limit of zero interlayer coupling, we must
have
w
(E;:)
)
+
0.
The Raman spectrum
(300
cm-'
I
w
I
3300
cm-')
for highly oriented pyrolytic graphite
(HOPG)'
is
shown in Fig. 2a, together with spectra (Fig. 2b-e) for
several other forms of
sp2
bonded carbons with vary-
ing degrees of intralayer and interlayer disorder.
For
HOPG,
a
sharp first-order line
at
1582
cm-' is ob-
served, corresponding
to
the Raman-active
E;:)
mode
observed
in
single crystal graphite at the same fre-
quency[3
I].
The first- and second-order mode fre-
quencies
of
graphite, disordered
sp2
carbons and
carbon nanotubes, are collected in Table
1.
Graphite exhibits strong second-order Raman-
active features. These features are expected and ob-
served in carbon tubules, as well. Momentum and
energy conservation, and the phonon density of states
determine,
to
a large extent, the second-order spectra.
By conservation
of
energy:
Aw
=
Awl
+
hw,,
where
o
and
wi
(i
=
1,2)
are, respectively, the frequencies
of
the incoming photon and those of the simultaneously
excited normal modes. There is
also
a crystal momen-
tum selection rule:
hk
=
Aq,
+
Aq,,
where
k
and
qi
(i
=
1.2) are, respectively, the wavevectors of the in-
coming photon and the two simultaneously excited
normal modes. Because
k
<<
qe,
where
qB
is a typical
wavevector on the boundary of the
BZ,
it follows that
ql
=
-q2.
For
a
second-order process, the strength of
the
IR
lattice absorption or Raman scattering is pro-
portional
to
IM(w)12g2(o),
where
g2(w)
=
gl(wl).
g,
(a,)
is
the two-phonon density of states subject to
the condition that
q1
=
-q2,
and where
g,
(w)
is
the
one-phonon density
of
states and
IM(w)I2
is the ef-
fective two-phonon Raman matrix element. In cova-
lently bonded solids, the second-order spectra1 features
are generally broad, consistent with the strong disper-
sion
(or
wide bandwidth) of both the optical and
acoustic phonon branches.
However, in graphite, consistent with the weak in-
terlayer interaction, the phonon dispersion parallel
to
(2)
'HOPG
is
a
synthetic polycrystalline form of graphite
produced
by
Union
Carbide[30].
The
c-axes
of
each
grain
(dia;
-1
pm)
are
aligned to
-1".
Fig.
2.
Raman spectra
(T
=
300
K)
from various
sp2
car-
bons
using
Ar-ion laser
excitation:
(a)
highly
ordered
pyro-
lytic
graphite
(HOPG),
(b)
boron-doped pyrolytic graphite
(BHOPG),
(c)
carbon nanoparticles (dia.
20
nm)
derived
from
the
pyrolysis
of
benzene
and graphitized
at
282OoC,
(d) as-synthesized carbon nanoparticles
(-85OoC),
(e)
glassy
carbon
(after
ref.
[24]).
the c-axis (i.e., along the
k,
direction) is small.
Also,
there is little in-plane dispersion of the optic branches
and acoustic branches near the zone corners and edges
(M
to
K).
This low dispersion enhances the peaks
in the one-phonon density of states,
g,
(w)
(Fig. la).
Therefore, relatively sharp second-order features are
observed in the Raman spectrum
of
graphite, which
correspond to characteristic combination
(wl
+
w2)
and overtone
(2w)
frequencies associated with these
low-dispersion (high one-phonon density of states) re-
gions in the
BZ.
For example,
a
second-order Raman
feature
is
detected at
3248
cm-', which is close to
2(1582 cm-')
=
3164
cm-', but significantly upshifted
due to the
3D
dispersion of the uppermost phonon
branch in graphite. The most prominent feature in
the graphite second-order spectrum is
a
peak close
to
2(1360
cm-')
=
2720
cm-' with
a
shoulder at
2698
cm-'
,
where the lineshape reflects the density
of
two-
phonon states in
3D
graphite. Similarly, for a
2D
graphene sheet, in-plane dispersion (Fig. Ib)
of
the
optic branches at the zone center and in the acoustic
132
P.
C.
EKLUND
et
ul.
Table
1.
Table of frequencies for graphitic carbons and nanotubes
Mode Planar graphite
assignment
*
t
(tube dia.)
HOPG[31]
BHOPG[31]
42'
127h
86Sg
-9mc
870' -900'
1582' 1585'
1577= 1591e
158Sg
1350' 1367'
1365e 1380'
1620'
2441' 2450'
2440e
2722' 2122c
2746e 2153e
2950'
2974e
3247' 3240'
3246e 3242e
Single-wall tubules Nested tubules
Holden$ Holden Chandrabhas Bacsa Kastner
et
ul.
[27]
et
ul.
[28]
Hiura
et
ul.
[24]
et
al.
[26]
et
ul.
[25]
(1-2
nm)
(1-2
nm)
et
ul.[23]
(15-50
nm)
(8-30
nm)
(20-80
nm)
-
1
566c'd
1592C,d
2681C*d
3
1
1568'
1594'
1341'
2450'
2680'
2925'
3180'
49,
58'
-700'
86V
1574' 1583' 158Ia 1582e
1575g
1340a
1353'
1356a
variesf
1620a
24Sa 2455' 2450' 2455e
2687' 2709' 2734'
2925e
3250a 3250a 3252=
*Activity: R
=
Raman-active, ir
=
infrared-active,
S
=
optically silent, observed in neutron scattering.
?Carbon atom displacement
II
or
I
to
e.
$Peaks in "difference spectrum" (see section
4.3).
a-eExcitation wavelength:
a742
nm,
b532
nm,
'514
nm,
d488
nm,
"458
nm;
absorption study; hfrom neutron scattering; 'predicted.
resonance Raman scattering study;
5r-
branches near the zone comers and edges is weak, giv-
ing rise to peaks in the one-phonon density of states.
One anticipates, therefore, that similar second-order
features will also be observed in carbon nanotubes.
This is because the zone folding (c.f., section
4)
pre-
serves in the tubule the essential character of the in-
plane dispersion of a graphene sheet for
q
parallel
to the tube axis. However, in small-diameter carbon
nanotubes, the cyclic boundary conditions around the
tube wall activate many new first-order Raman- and
IR-active modes, as discussed below.
Figure 2b shows the Raman spectrum of Boron-
doped, highly oriented pyrolytic-graphite (BHOPG)
according to Wang
et.
aZ[32]. Although the
BHOPG
spectrum is similar to that of HOPG, the effect of the
0.5"/0
substitutional boron doping is to create in-plane
disorder, without disrupting the overall
AB
stacking
of
the layers or the honeycomb arrangement of the re-
maining C-atoms in the graphitic planes. However, the
boron doping relaxes the
q
=
0
optical selection rule
for single-phonon scattering, enhancing the Raman ac-
tivity
of
the graphitic one- and two-phonon density
of states. Values for the peak frequencies
of
the first-
and second-order bands in
BHOPG
are tabulated in
Table
1.
Significant disorder-induced Raman activity
in the graphitic one-phonon density
of
states is
ob-
served near 1367 cm-', similar
to
that observed in
other disordered
sp2
bonded carbons, where features
in the range -1360-1365 cm-' are detected. This
band is referred to in the literature as the "D-band,"
and the position of this band has been shown to de-
pend weakly on the laser excitation wavelength[32].
This
unusual effect arises from
a
resonant coupling of
the excitation laser with electronic states associated
with the disordered graphitic material. Small basal
plane crystallite size
(L,)
has also been shown[33]
to
activate disorder-induced scattering in the D-band.
The high frequency
E$:)(
q
=
0)
mode has also been
investigated in a wide variety of graphitic materials
that have various degrees
of
in-plane and stacking dis-
order[32], The frequency, strength, and line-width of
this mode is also found to be a function
of
the degree
of
the disorder, but the peak position depends much
less strongly on the excitation frequency.
The Raman spectrum of a strongly disordered
sp2
carbon material, "glassy" carbon, is shown in Fig. 2e.
The Eii'-derived band is observed at
1600
cm-' and
is broadened along with the D-band at
1359
cm-'.
The similarity
of
the spectrum
of
glassy carbon (Fig. 2e)
to
the one-phonon density
of
states
of
graphite (Fig.
la)
is apparent, indicating that despite the disorder, there
is still
a
significant degree of
sp2
short-range order in
the glassy carbon. The strongest second-order feature
is
located
at
2973 cm-', near
a
combination band
(wl
+
w2)
expected in graphite
at
D (1359
m-I)
+
E'
(1620
cm-')
=
2979 cm-', where the
Eig
(1620
cm
I)
frequency is associated with
a
mid-zone max-
imum
of
the uppermost optical branch in graphite
(Fig. la).
The carbon black studied here was prepared
by
a
C02
laser-driven pyrolysis of
a
mixture of benzene,
ethylene, and iron carbonyI[34].
As
synthesized,
TEM
2g-
Vibrational modes
of
carbon nanotubes
133
images show that this carbon nanosoot consists of dis-
ordered
sp2
carbon particles with an average particle
diameter of -200
A.
The Raman spectrum (Fig. 2d)
of the “as synthesized” carbon black is very similar
to
that of glassy carbon (Fig.
2e)
and has broad disorder-
induced peaks in the first-order Raman spectrum
at
1359
and 1600 cm-’, and
a
broad second-order fea-
ture near 2950 cm-’. Additional weak features are
observed in the second-order spectrum at 2711 and
3200 cm-’
,
similar
to
values
in
HOPG,
but appear-
ing closer
to
2(1359
CII-’)
=
2718 cm-’ and 2(1600
cm-I)
=
3200 cm-’
,
indicative
of
somewhat weaker
3D
phonon dispersion, perhaps due to weaker cou-
pling between planes in the nanoparticles than found
in
HQPG.
TEM
images[34] show that the heat treat-
ment
of
the laser pyrolysis-derived carbon nanosoot
to
a
temperature
THT
=
2820°C graphitizes the
nano-
particles (Le., carbon layers spaced by -3.5
A
are
aligned parallel
to
facets on hollow polygonal parti-
cles). As indicated in Fig. 2c, the Raman spectrum of
this heat-treated carbon black
is
much more “gra-
phitic” (similar to Fig. 2a) and, therefore, a decrease
in
the integrated intensity of the disorder-induced band
at
1360 cm-’ and
a
narrowing of the 1580 cm-’ band
is observed. Note that heat treatment allows
a
shoul-
der associated with a maximum in the mid-BZ density
of
states
to
be resolved at 1620 cm-I, and dramati-
cally enhances and sharpens the second-order features.
3.
THEORY
OF
VIBRATIONS IN
CARBON NANOTUBFS
A
single-,wall carbon nanotube can be visualized by
referring
to
Fig. 3, which shows a 2D graphene sheet
with lattice vectors
a1
and
a2,
and
a
vector
C
given by
where
n
and
m
are integers. By rolling the sheet such
that the tip and tail
of
C
coincide, a cylindrical nano-
Fig.
3.
Translation
vectors
used
to
define
the symmetry
of
a
carbon
nanotube
(see text).
The
vectors
a,
and
a2
define
the
2D
primitive
cell.
tube specified by
(n,
m)
is
obtained.
If
n
=
m,
the re-
sulting nanotube is referred to as an “armchair”
tubule, while if
n
=
0
or
m
=
0,
it is referred to as a
“zigzag” tubule; otherwise
(n
#
m
#
0)
it is known as
a
“chiral” tubule. There is no loss of generality if it
is
assumed that
n
>
m.
The electronic properties of single-walled carbon
nanotubes have been studied theoretically using dif-
ferent methods[4-121.
It
is found that if
n
-
m
is a
multiple
of
3, the nanotube will be metallic; otherwise,
it
will
exhibit a semiconducting behavior. Calculations
on
a
2D array
of
identical armchair nanotubes with
parallel tube axes within the local density approxi-
mation framework indicate that a crystal with a hex-
agonal packing of the tubes is most stable, and that
intertubule interactions render the system semicon-
ducting with
a
zero energy gap[35].
3.1
Symmetry
groups
of
nanotubes
A
cylindrical carbon nanotube, specified by
(n,m),
can be considered a one-dimensional crystal with a
fundamental lattice vector
T,
along the direction of the
tube axis, of length given by[1,3]
where
dR
=
d
if
n
-
rn
#
3dr
=
3d if
n
-
m
=
3dr (3)
where Cis the length
of
the vector in eqn
(l),
d
is the
greatest common divisor of
n
and
m,
and
r
is
any in-
teger. The number
of
atoms per unit cell
is
2N such
that
N
=
2(n2
+
m2
+
nm)/dR.
(4)
For a chiral nanotube specified by
(n,
m),
the cylin-
der is divided into
d
identical sections; consequently
a
rotation about the tube
axis
by the angle 2u/d con-
stitutes
a
symmetry operation. Another symmetry op-
eration,
R
=
($,
7)
consists
of
a
rotation by an angle
$
given by
s2
$=27r-
Nd
followed by a translation
7,
along the direction of the
tube axis, given by
d
N
s=T-.
The quantity
s2
that appears in eqn (5)
is
expressed in
terms of
n
and
m
by the relation
s2
=
(p(m
+
2n)
+
q(n
+
2m)I
(d/dR)
(7)
134
P.
C.
EKLUND
et
ai.
where
p
and
q
are integers that are uniquely deter-
mined by the eqn
rnp
-
nq
=
d,
(8)
subject to the conditions
q
<
m/d
and
p
<
n/d.
For the case
d
=
1,
the symmetry group of
a
chi-
ral nanotube specified by
(n,
m)
is a cyclic group
of
order
N
given by
where
E
is the identity element, and
(RNjn
=
(2~
(WN),
T/N))
.
For the general case when
d
#
1,
the cylinder
is divided into
d
equivalent sections. Consequently, it
follows that the symmetry group of the nanotube is
given by
where
and
Here the operation
ed
represents a rotation by
2n/d
about the tube axis; the angles
of
rotation in
(!?hd/fi
are defined modulo
2?r/d,
and the symmetry element
The irreducible representations
of
the symmetry
group
C?
are given by
A,
B,
El,
E2,
. .
.
,
EN/Z-,
.
The
A
representation is completely symmetric, while in the
B
representation, the characters for the operations
cd
and
6iNd/Q
are
(RNd/n
=
(2~(fi/Nd),Td/N)).
and
In the
E,
irreducible representation, the character of
any symmetry operation corresponding to
a
rotation
by an angle is given by
Equations
(13-15)
completely determine the character
table of the symmetry group
e
for
a
chiral nanotube.
Applying the above symmetry formulation
to
arm-
chair
(n
=
m)
and zigzag
(m
=
0)
nanotubes, we find
that such nanotubes have
a
symmetry group given by
the product of the cyclic group
e,
and where
e;,
consists
of
only
two
symmetry operations: the
identity, and a rotation by
21r/2n
about the tube axis
followed by a translation by
T/2.
Armchair and zig-
zag nanotubes, however, have other symmetry oper-
ations, such as inversion and reflection in planes
parallel to the tube axis. Thus, the symmetry group,
assuming an infinitely long nanotube with
no
caps,
is
given by
Thus
e
=
6)2,h
in these cases. The choice of
a,,,
or
Dnh in eqn
(16)
is made to insure that inversion is a
symmetry operation of the nanotube. Even though we
neglect the caps in calculating the vibrational frequen-
cies, their existence, nevertheless, reduces the symme-
try
to
either
Bnd
or
Bnh.
Of course, whether the symmetry groups for arm-
chair and zigzag tubules are taken to be
d)&
(or
a,,,)
or
a)2nh,
the calculated vibrational frequencies will be
the same; the symmetry assignments for these modes,
however, will be different. It is, thus, expected that
modes that are Raman or IR-active under
and
or
TInh
but are optically silent under
BZnh
will only show
a
weak activity resulting from the fact that the existence
of
caps lowers the symmetry that would exist for
a
nanotube
of
infinite length.
3.2
Model
calculations
of
phonon
modes
The BZ of a nanotube is a line segment along the
tube direction, of length
2a/T.
The rectangle formed
by vectors
C
and
T,
in Fig.
3,
has an area
N
times
larger than the area
of
the unit cell of a graphene sheet
formed by vectors
al
and
a2,
and gives rise to
a
rect-
angular BZ than is Ntimes smaller than the hexago-
nal BZ
of
a graphene sheet. Approximate values for
the vibrational frequencies of the nanotubes can be
obtained from those of a graphene sheet by the
method of zone folding, which in this case implies that
In the above eqn,
1D
refers to the nanotubes whereas
2D
refers to the graphene sheet,
k
is the
1D
wave vec-
tor, and ?and
e
are
unit
vectors along the tubule
axis
and vector
C,
respectively,
and
p
labels the tubule pho-
non
branch.
The phonon frequencies of a
2D
graphene sheet,
for carbon displacements both parallel and perpendic-
ular to the sheet, are obtained[l] using
a
Born-Von
Karman model similar to that applied successfully
to
3D
graphite. C-C interactions up
to
the fourth nearest
in-plane neighbors were included. For
a
2D
graphene
sheet, starting from the previously published force-
constant model of
3D
graphite, we set all the force
constants connecting atoms
in
adjacent layers to zero,
and we modified the in-plane force constants slightly
to
describe accurately the results
of
electron energy
loss
Vibrational
modes
of
carbon nanotubes
135
spectroscopic measurements, which yield the phonon
dispersion curves along the
M
direction in the BZ. The
dispersion curves are somewhat different near
M,
and
along
M-K,
than the 2D calculations shown in Fig. Ib.
The lattice dynamical model for 3D graphite produces
dispersion curves
q(q)
that are in good agreement
with experimental results from inelastic neutron scat-
tering, Raman scattering, and IR spectroscopy.
The zone-folding scheme has two shortcomings.
First, in
a
2D graphene sheet, there are three modes
with vanishing frequencies as
q
+
0;
they correspond
to
two translational modes with in-plane C-atom dis-
placements and one mode with out-of-plane C-atom
displacements. Upon rolling the sheet into a cylinder,
the translational mode in which atoms move perpen-
dicular
to
the plane will now correspond to the breath-
ing mode of the cylinder for which the atoms vibrate
along the radial direction. This breathing mode has a
nonzero frequency, but the value cannot be obtained
by zone folding; rather, it must be calculated analyt-
ically. The frequency of the breathing mode
w,,d
is
readily calculated and
is
found
to
be[l,2]
where
a
=
2.46
A
is
the lattice constant of a graphene
sheet,
ro
is the tubule radius,
mc
is the mass
of
a car-
bon atom, and
+:)
is
the bond stretching force con-
stant between an atom and its
ith
nearest neighbor. It
should be noted that the breathing mode frequency is
found
to
be independent of
n
and
m,
and that it is in-
versely proportional
to
the tubule radius. The value
of
=
300
cmp' for
r,
=
3.5
A,
the radius that cor-
responds to a nanotube capped by a
C60
hemisphere.
Second, the zone-folding scheme cannot give rise
to the two zero-frequency tubule modes that corre-
spond to the translational motion of the atoms in the
two directions perpendicular to the tubule axis. That
is
to
say, there are
no
normal modes in the
2D
graph-
ene sheet for which the atomic displacements are such
that if the sheet is rolled into
a
cylinder, these displace-
ments would then correspond to either of the rigid tu-
bule translations in the directions perpendicular to the
cylinder
axis.
To
convert these two translational modes
into eigenvectors of the tubule dynamical matrix, a
perturbation matrix must be added to the dynamical
matrix.
As
will be discussed later, these translational
modes transform according
to
the
El
irreducible rep-
resentation; consequently, the perturbation should be
constructed
so
that it will cause a mixing of the
El
modes, but should have no effect in first order on
modes with other symmetries. The perturbation ma-
trix turns out
to
cause the frequencies
of
the
El
modes with lowest frequency to vanish, affecting the
other
El
modes only slightly.
Finally, it should be noted that in the
zone-folding
scheme, the effect
of
curvature on the force constants
has been neglected. We make this approximation un-
der the assumption that the hybridization between the
sp2
and
pz
orbitals is small. For example, in the arm-
chair nanotube based on
CG0,
with a diameter of ap-
proximately
0.7
nm, the three bond angles are readily
calculated and they are found to be
120.00",
118.35',
and
118.35'.
Because the deviation of these angles
from 120" is very small, the effect
of
curvature on the
force constants might be expected to be small. Based
on
a calculation using the semi-empirical interatomic
Tersoff potential, Bacsa
et
al.
[26,36] estimate consid-
erable mode softening with decreasing diameter. For
tubes of diameter greater than
-10
nm,
however, they
predict tube wall curvature has negligible effect on the
mode frequencies.
3.3
Raman- and infrared-active modes
The frequencies of the tubule phonon modes at the
r-point, or BZ center, are obtained from eqn
(17)
by
setting
k
=
0.
At this point, we can classify the modes
according to the irreducible representations of the
symmetry group that describes the nanotube. We be-
gin by showing how the classification works in the case
of chiral tubules. The nanotube modes obtained from
the zone-folding eqn by setting
p
=
0
correspond to
t-he I'-point modes
of
the
2D
graphene sheet. For these
modes, atoms connected by any lattice vector
of
the
2D sheet have the same displacement. Such atoms, un-
der the symmetry operations of the nanotubes, trans-
form into each other; consequently, the nanotubes
modes obtained by setting
1.1
=
0
are completely sym-
metric and they transform according to the
A
irreduc-
ible representation.
Next, we consider the r-point nanotube modes ob-
tained by setting
k
=
0
and
p
=
N/2
in eqn
(17).
The
modes correspond
to
2D
graphene sheet modes at the
point
k
=
(Mr/C)e
in the hexagonal BZ. We consider
how such modes transform under the symmetry op-
erations of the groups
ed
and
C3hd/,.
Under the ac-
tion of the symmetry element C,, an atom in the
2D
graphene sheet is carried into another atom separated
from it by the vector
The displacements
of
two such atoms at the point
k
=
(Nr/C)C
have a phase difference given by
N
2
-
k.rl
=
27r(n2
+
m2
+
nm)/(dciR)
(20)
which is an integral multiple
of
2n.
Thus,
the displace-
ments of the two atoms are equal and it follows that
The symmetry operation
RNd,,
carries an atom
into another
one
separated from it by the vector
136
P.
c.
EKI
wherep and
q
are the integers uniquely determined by
eqn
(8).
The atoms in the
2D
graphene sheet have dis-
placements,
at
the point
k
=
(Nr/C)&,
that are com-
pletely out of phase. This follows from the observation
that
and that
Wd
is an odd integer; consequently
From the above, we therefore conclude that the nano-
tube modes obtained by setting
p
=
N/2,
transform
according to the
B
irreducible representation
of
the
chiral symmetry group
e.
Similarly, it can be shown that the nanotube modes
at the I?-point obtained from the zone-folding eqn by
setting
p
=
9,
where
0
<
9
<
N/2,
transform accord-
ing to the
Ev
irreducible representation of the symme-
try group
e.
Thus, the vibrational modes at the
F-point
of
a chiral nanotube can be decomposed ac-
cording to the following eqn
Modes with
A,
E,,
or
E2
symmetry are Raman ac-
tive, while only
A
and
El
modes are infrared active.
The
A
modes are nondegenerate and the
E
modes are
doubly degenerate. According to the discussion in the
previous section, two
A
modes and one of the
E,
modes have vanishing frequencies; consequently, for
a
chiral nanotube there are
15
Raman- and
9
IR-active
modes, the IR-active modes being also Raman-active.
It should be noted that the number
of
Raman- and IR-
active modes is independent
of
the nanotube diameter.
For
a
given chirality,
as
the diameter of the nanotube
increases, the number of phonon modes
at
the
BZ
cen-
ter also increases. Nevertheless, the number
of
the
modes that transform according to the
A,
E,,
or
E2
irreducible representations does not change. Since only
modes with these symmetries will exhibit optical activ-
ity, the number of Raman
or
IR modes does not in-
crease with increasing diameter. This, perhaps unantic-
ipated, result greatly simplifies the data analysis. The
symmetry classification
of
the phonon modes in arm-
chair and zigzag tubules have been studied in ref.
[2,3]
under the assumption that the symmetry group of
these tubules is isomorphic with either
Dnd
or
Bnh,
depending on whether
n
is odd or even.
As
noted ear-
lier,
if
one considers an infinite tubule with no caps,
the relevant symmetry group for armchair and zigzag
tubules would be the group
6)2nh.
For armchair tu-
bules described by the
Dnd
group there are, among
others,
3A1,,
6E1,, 6E2,,
2A2,,
and
SEI,
optically
active modes with nonzero frequencies; consequently,
there are
15
Raman- and
7
IR-active modes. All zig-
zag tubules, under
Dnd
or
Bnh
symmetry group have,
among others,
3A1,,
6E,,, 6E2,,
2A2,,
and
5E,,
op-
:UND
et
al.
tically active modes with nonzero frequencies; thus
there are
15
Raman- and
7
IR-active modes.
3.4
Mode frequency dependence
on
tubule diameter
In Figs.
4-6,
we display the calculated tubule fre-
quencies as
a
function
of
tubule diameter. The results
are based on the zone-folding model
of
a
2D
graph-
ene sheet, discussed above. IR-active
(a)
and Raman-
active (b) modes appear separately for chiral tubules
(Fig.
4),
armchair tubules (Fig.
5)
and zig-zag tubules
(Fig.
6).
For the chiral tubules, results for the repre-
sentative
(n,
m),
indicated
to
the left in the figure, are
displaced vertically according
to
their calculated diam-
eter, which is indicated
on
the right. Similar
to
modes
in a
Ca
molecule, the lower and higher frequency
modes are expected, respectively, to have radial and
tangential character. By comparison of the model cal-
culation results in Figs. 4-6 for the three tube types
(armchair, chiral, and zig-zag) a common general be-
havior is observed for both the IR-active (a) and
Raman-active (b) modes. The highest frequency
modes exhibit much less frequency dependence on di-
ameter than the lowest frequency modes. Taking the
large-diameter tube frequencies as our reference, we
see that the four lowest modes stiffen dramatically
(150-400
cm-') as the tube diameter approaches
-1
nm. Conversely, the modes above
-800
cm-' in the
large-diameter tubules are seen
to
be relatively less sen-
sitive to tube diameter: one Raman-active mode stiff-
ens with increasing tubule diameter (armchair), and a
few modes in all the three tube types soften
(100-200
cm-'), with decreasing tube diameter. It should also
be noted that, in contrast to armchair and zig-zag tu-
bules, the mode frequencies in chiral tubules are
grouped near
850
cm-' and
1590
cm-'.
All carbon nanotube samples studied to date have
been undoubtedly composed
of
tubules with a distri-
bution
of
diameters and chiralities. Therefore, whether
one is referring
to
nanotube samples comprised
of
single-wall tubules or nested tubules, the results in
Figs.
4-6
indicate one should expect inhomogeneous
broadening
of
the IR- and Raman-active bands, par-
ticularly if the range of tube diameter encompasses the
1-2
nm range. Nested tubule samples must have
a
broad diameter distribution and,
so,
they should ex-
hibit broader spectral features due to inhomogeneous
broadening.
4.
SYNTHESIS AND RAMAN SPECTROSCOPY
OF
CARBON NANOTUBES
We next address selected Raman scattering data
collected on nanotubes, both in our laboratory and
elsewhere. The particular method
of
tubule synthesis
may also produce other carbonceous matter that is
both difficult
to
separate from the tubules and also ex-
hibits potentially interfering spectral features. With
this in mind, we first digress briefly
to
discuss synthe-
sis and purification techniques used
to
prepare nano-
tube samples.
Vibrational modes
of
carbon nanotubes
(32.12)
(28,16)
(24,9)
137
IRI
I
11111
I
I
111
I
I
I
I
1
I I I I I
,
II
I
II
I
I
II
II
I
I
U
I
I1
I
I
II
/Ill
I
I1
ni
Ill1
II
I
I111
11111
I
I
I
I
I
L-L
0
400
800 1200
1600
Frequency
(cm-l)
(a)
I
I I
I I I
I I
I
I1
111
IR
I
II
I
I1
I
II
I
II
I
I1
I
I
II
II
I
I
n
I
II
II
I
II
111
31.0
30.4
n
23.3
22.8
~
0
15.5
15.2
-2
1.12
1.55
Y
n
31.0
30.4
23.3
0s
22.8
n
h
Q)
u
i
n
15.5
15.2
.z
7.72
1.55
I I I I
I
I
I
I
400
800
1200 1600
Frequency
(cm-I)
(b)
Fig.
4.
Diameter dependence
of
the first
order
(a) IR-active and,
(b)
Raman-active mode frequencies for
“chiral” nanotubes.
4.1
Synthesis and purification
Nested carbon nanotubes, consisting
of
closed con-
centric, cylindrical tubes were first observed by Iijima
by TEM[37]. Later TEM studies[38] showed that the
tubule ends were capped by the inclusion
of
pentagons
and that the tube walls were separated by
-3.4
A.
A
dc carbon-arc discharge technique for large-scale
syn-
thesis of nested nanotubes was subsequently reported
[39].
In
this technique,
a
dc arc is struck between two
graphite electrodes under an inert helium atmosphere,
as
is
done
in
fullerene generation. Carbon vaporized
from the anode condenses on the cathode to form a
hard, glassy outer core
of
fused carbon and
a
soft,
black inner core containing
a
high concentration
of
nanotubes and nanoparticles. Each nanotube typically
contains between
10
and
100
concentric tubes that are
grouped in “microbundles” oriented axially within the
core[l4].
These nested nanotubes may be harvested from the
core by grinding and sonication; nevertheless, substan-
tial fractions
of
other types
of
carbon remain, all of
which are capable
of
producing strong Raman bands
as
discussed in section
2.
It is very desirable, therefore,
to remove as much
of
these impurity carbon phases
as
possible. Successful purification schemes that exploit
the greater oxidation resistance of carbon nanotubes
have been investigated
[40-421.
Thermogravimetric
analyses reveal weight
loss
rate maxima at
420”C,
585°C)
and
645°C
associated with oxidation (in air)
of fullerenes, amorphous carbon soot, and graphite,
respectively,
to
form volatile
CO
and/or
COz.
Nano-
tubes and onion-like nanoparticles were found to lose
weight rapidly at higher temperatures around 695°C.
Evidently, the concentration of these other forms
of
carbon can be lowered by oxidation. However, the
abundant carbon nanoparticles, which are expected
to
have a Raman spectrum similar to that shown
in
Figs.
Id
or
IC
are more difficult to remove in this way. Never-
theless, Ebbesen
et
al.
[43] found that, by heating core
material to
700°C
in air until more than
99%
of the
starting material had been removed by oxidation, the
remaining material consisted solely
of
open-ended,
nested nanotubes. The oxidation was found to initiate
at the reactive end caps and progress toward the cen-